

A262114


Irregular triangle read by rows: row b (b >= 2) gives periodic part of digits of the baseb expansion of 1/5.


2



0, 0, 1, 1, 0, 1, 2, 1, 0, 3, 1, 1, 1, 2, 5, 4, 1, 4, 6, 3, 1, 7, 2, 2, 2, 4, 9, 7, 2, 7, 10, 5, 2, 11, 3, 3, 3, 6, 13, 10, 3, 10, 14, 7, 3, 15, 4, 4, 4, 8, 17, 13, 4, 13, 18, 9, 4, 19, 5, 5, 5, 10, 21, 16, 5, 16, 22, 11, 5, 23, 6, 6, 6, 12, 25, 19, 6, 19, 26, 13, 6, 27, 7, 7, 7, 14, 29, 22, 7, 22
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OFFSET

2,7


COMMENTS

The number of terms associated with a particular value of b are cyclical: 4, 4, 2, 1, 1, repeat. This is because the values are associated with b (mod 5), starting with 2 (mod 5).
The expansion of 1/5 either terminates after one digit when b == 0 (mod 5) or is purely recurrent in all other cases of b (mod 5), since 5 is prime and must either divide or be coprime to b.
The period for purely recurrent expansions of 1/5 must be a divisor of Euler's totient of 5 = 4, i.e., one of {1, 2, 4}.
b == 0 (mod 5): 1 (terminating)
b == 1 (mod 5): 1 (purely recurrent)
b == 2 (mod 5): 4 (purely recurrent)
b == 3 (mod 5): 4 (purely recurrent)
b == 4 (mod 5): 2 (purely recurrent)
The expansion of 1/5 has a fulllength period 4 when base b is a primitive root of p = 5.
Digits of 1/5 for the following bases:
2 0, 0, 1, 1
3 0, 1, 2, 1
4 0, 3
5* 1
6 1
7 1, 2, 5, 4
8 1, 4, 6, 3
9 1, 7
10* 2
11 2
12 2, 4, 9, 7
13 2, 7, 10, 5
14 2, 11
15* 3
16 3
17 3, 6, 13, 10
18 3, 10, 14, 7
19 3, 15
20* 4
...
Asterisks above denote terminating expansion; all other entries are digits of purely recurrent reptends.
Each entry associated with base b with more than one term has a second term greater than the first except for b = 2, where the first two terms are 0, 0.
Entries for b == 0 (mod 5) (i.e., integer multiples of 5) appear at 11, 23, 35, ..., every 12th term thereafter.


REFERENCES

U. Dudley, Elementary Number Theory, 2nd ed., Dover, 2008, pp. 119126.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 6th ed., Oxford Univ. Press, 2008, pp. 138148.
Oystein Ore, Number Theory and Its History, Dover, 1988, pp. 311325.


LINKS

Michael De Vlieger, Table of n, a(n) for n = 2..10000
Eric Weisstein's World of Mathematics, Decimal Period.
Eric Weisstein's World of Mathematics, Repeating Decimal.


FORMULA

Conjectures from Colin Barker, Oct 09 2015: (Start)
a(n) = 2*a(n12)  a(n24) for n>24.
G.f.: x^3*(x^19 +x^18 +x^17 +2*x^16 +2*x^15 +x^14 +2*x^13 +3*x^12 +2*x^11 +x^10 +x^9 +x^8 +3*x^7 +x^5 +2*x^4 +x^3 +x +1) / (x^24 2*x^12 +1).
(End)


EXAMPLE

For b = 8, 1/5 = .14631463..., thus 1, 4, 6, 3 are terms in the sequence.
For b = 10, 1/5 = .2, thus 2 is a term in the sequence.
For b = 13, 1/5 = .27a527a5..., thus 2, 7, 10, 5 are terms in the sequence.


MATHEMATICA

RotateLeft[Most@ #, Last@ #] &@ Flatten@ RealDigits[1/5, #] & /@ Range[2, 38] // Flatten (* Michael De Vlieger, Sep 11 2015 *)


CROSSREFS

Cf. A004526 Digits of expansions of 1/2.
Cf. A026741 Full Reptends of 1/3.
Cf. A130845 Digits of expansions of 1/3 (eliding first 2 terms).
Cf. A262115 Digits of expansions of 1/7.
Sequence in context: A176808 A327029 A167192 * A320780 A029312 A287352
Adjacent sequences: A262111 A262112 A262113 * A262115 A262116 A262117


KEYWORD

nonn,base,tabf


AUTHOR

Michael De Vlieger, Sep 11 2015


STATUS

approved



